Models of the quark mixing matrix

The structure of the Cabibbo-Kobayashi-Maskawa (CKM) matrix is analyzed from the standpoint of illustrative composite models. It is shown how electroweak couplings can yield information on substructure. Models are constructed with two and three families of quarks, by taking tensor products of sufficient numbers of spin-½ representations and imagining the dominant terms in the mass matrix to arise from spin-spin interactions. Assumptions are made about the absence of certain terms. Generic results then obtained include the familiar relation $\left|V_{\mathrm{us}}\right|={\left(\frac{m_d}{m_s}\right)}^{\frac{1}{2}}-{\left(\frac{m_u}{m_c}\right)}^{\frac{1}{2}}$, and a less frequently seen relation $\left|V_{\mathrm{cb}}\right|=\sqrt{2}\left[\right(\frac{m_s}{m_b})-(\frac{m_c}{m_t}\left)\right]$. The magnitudes of $V_{\mathrm{ub}}$ and $V_{\mathrm{td}}$ come out naturally to be of the right order. The phase in the CKM matrix can be put in by hand, but its origin remains obscure.